Unitary part of a contraction

For a contraction A on a Hilbert space H, we define the index j (A) (resp., k (A)) as the smallest nonnegative integer j (resp., k) such that ker (I - A^{j *} A^j) (resp., ker (I - A^{k *} A^k) ∩ ker (I - A^k A^{k *})) equals the subspace of H on which the unitary part of A acts. We show that if n = dim H < ∞, then j (A) ≤ n (resp., k (A) ≤ ⌈ n / 2 ⌉), and the equality holds if and only if A is of class S_n (resp., one of the three conditions is true: (1) A is of class S_n, (2) n is even and A is completely nonunitary with {norm of matrix} A^{n - 2} {norm of matrix} = 1 and {norm of matrix} A^{n - 1} {norm of matrix} < 1, and (3) n is even and A = U ⊕ A^′, where U is unitary on a one-dimensional space and A^′ is of class S_{n - 1}).